As you say, the theory of "complex multiplication" (looking at modular invariants of elliptic curves with CM) solves the problem of determining the maximal abelian extension of an imaginary quadratic field. But as yet no analogous theory is known for real quadratic fields, although class field theory affords  a complete theoretical (but not explicit) description of the abelian extensions of a given number field. You can  take the measure of our ignorance by  listening to the short negative answer given by Serre at the end of his talk on abelian Galois theory at the occasion of Galois' bicentenary (the talk can be found on You Tube, I think).